This paper presents a distributed algorithm for wireless adhoc networks that runs in polylogarithmic number of rounds in the size of the network and constructs a lightweight, linear size, (1 + ε)-spanner for any given ε > 0. A wireless network is modeled by a d-dimensional α-quasi unit ball graph (α-UBG), which is a higher dimensional generalization of the standard unit disk graph (UDG) model. The d-dimensional α-UBG model goes beyond the unrealistic "flat world" assumption of UDGs and also takes into account transmission errors, fading signal strength, and physical obstructions. The main result in the paper is this: for any fixed ε > 0, 0 < α ≤ 1, and d ≥ 2 there is a distributed algorithm running in O(log n·log * n) communication rounds on an n-node, d-dimensional α-UBG G that computes a (1 + ε)-spanner G of G with maximum degree Δ(G ) = O(1) and total weight w(G ) = O(w(MST (G)). This result is motivated by the topology control problem in wireless ad-hoc networks and improves on existing topology control algorithms along several dimensions. The technical contributions of the paper include a new, sequential, greedy algorithm with relaxed edge ordering and lazy updating, and clustering techniques for filtering out unnecessary edges.

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We start by defining a pruning process involving sellers on one side and buyers on the other. The goal is to quickly select a subset of the sellers so that the products that these sellers bring to the market has small cost ratio, i.e., the ratio of the total cost of the selected sellers' products to amount that interested buyers are willing to pay. As modeled here, the pruning process can be used to speed up distributed implementations of greedy algorithms (e.g., for minimum dominating set, facility location, etc). We present a randomized instance of the pruning process that, for any positive k, runs in O(k) communication rounds with O(log N )-sized messages, yielding a cost ratio of O(N c/k ). Here N is the product of the number of sellers and number of buyers and c is a small constant. Using this O(k)-round pruning algorithm as the basis, we derive several simple, greedy, O(k)-round distributed approximation algorithms for MDS and facility location (both metric and non-metric versions). Our algorithms achieve optimal approximation ratios in polylogarithmic rounds and shave a "logarithmic factor" off the best, known, approximation factor, typically achieved using LProunding techniques.

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